# Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification

arXiv: https://arxiv.org/abs/1802.00850 · HTML: https://mlsanity.com/paper/1802.00850/

## Key facts

- Cited by 81 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Feb 2018.

## Built on by (most cited first, 30 of 81)

- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs](https://mlsanity.com/paper/2109.07143/) (2021)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [Bi-fidelity Modeling of Uncertain and Partially Unknown Systems using DeepONets](https://mlsanity.com/paper/2204.00997/) (2022)
- [Neural and spectral operator surrogates: unified construction and expression rate bounds](https://mlsanity.com/paper/2207.04950/) (2022)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems](https://mlsanity.com/paper/2202.04648/) (2022)
- [Transfer Learning on Multi-Fidelity Data](https://mlsanity.com/paper/2105.00856/) (2021)
- [Manifold learning-based polynomial chaos expansions for high-dimensional surrogate models](https://mlsanity.com/paper/2107.09814/) (2021)
- [Multi-fidelity wavelet neural operator with application to uncertainty quantification](https://mlsanity.com/paper/2208.05606/) (2022)
- [Adaptive operator learning for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2310.17844/) (2023)
- [Agnostic Active Learning of Single Index Models with Linear Sample Complexity](https://mlsanity.com/paper/2405.09312/) (2024)
- [Novel Deep neural networks for solving Bayesian statistical inverse](https://mlsanity.com/paper/2102.03974/) (2021)
- [TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems](https://mlsanity.com/paper/2105.12033/) (2021)
- [Physics-informed neural networks for solving parametric magnetostatic problems](https://mlsanity.com/paper/2202.04041/) (2022)
- [Scalable conditional deep inverse Rosenblatt transports using tensor-trains and gradient-based dimension reduction](https://mlsanity.com/paper/2106.04170/) (2021)
- [A Review of Physics-based Machine Learning in Civil Engineering](https://mlsanity.com/paper/2110.04600/) (2021)
- [Generative AI and Process Systems Engineering: The Next Frontier](https://mlsanity.com/paper/2402.10977/) (2024)
- [Bi-fidelity Variational Auto-encoder for Uncertainty Quantification](https://mlsanity.com/paper/2305.16530/) (2023)
- [Variational Inference with NoFAS: Normalizing Flow with Adaptive Surrogate for Computationally Expensive Models](https://mlsanity.com/paper/2108.12657/) (2021)
- [A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic](https://mlsanity.com/paper/2204.06362/) (2022)
- [Leveraging Industry 4.0 -- Deep Learning, Surrogate Model and Transfer Learning with Uncertainty Quantification Incorporated into Digital Twin for Nuclear System](https://mlsanity.com/paper/2210.00074/) (2022)
- [Conformal Approach To Gaussian Process Surrogate Evaluation With Coverage Guarantees](https://mlsanity.com/paper/2401.07733/) (2024)
- [Surrogate Learning in Meta-Black-Box Optimization: A Preliminary Study](https://mlsanity.com/paper/2503.18060/) (2025)
- [Deep Adaptive Arbitrary Polynomial Chaos Expansion: A Mini-data-driven Semi-supervised Method for Uncertainty Quantification](https://mlsanity.com/paper/2107.10428/) (2021)
- [slimTrain -- A Stochastic Approximation Method for Training Separable Deep Neural Networks](https://mlsanity.com/paper/2109.14002/) (2021)
- [Polynomial Chaos Expansions on Principal Geodesic Grassmannian Submanifolds for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/2401.16683/) (2024)
- [HiLAB: A Hybrid Inverse-Design Framework](https://mlsanity.com/paper/2505.17491/) (2025)
- [Surrogate and inverse modeling for two-phase flow in porous media via theory-guided convolutional neural network](https://mlsanity.com/paper/2110.10080/) (2021)
- [PCENet: High Dimensional Surrogate Modeling for Learning Uncertainty](https://mlsanity.com/paper/2202.05063/) (2022)

## Abstract (arXiv)

State-of-the-art computer codes for simulating real physical systems are often characterized by a vast number of input parameters. Performing uncertainty quantification (UQ) tasks with Monte Carlo (MC) methods is almost always infeasible because of the need to perform hundreds of thousands or even millions of forward model evaluations in order to obtain convergent statistics. One, thus, tries to construct a cheap-to-evaluate surrogate model to replace the forward model solver. For systems with large numbers of input parameters, one has to deal with the curse of dimensionality - the exponential increase in the volume of the input space, as the number of parameters increases linearly. In this work, we demonstrate the use of deep neural networks (DNN) to construct surrogate models for numerical simulators. We parameterize the structure of the DNN in a manner that lends the DNN surrogate the interpretation of recovering a low dimensional nonlinear manifold. The model response is a parameterized nonlinear function of the low dimensional projections of the input. We think of this low dimensional manifold as a nonlinear generalization of the notion of the active subspace. Our approach is demonstrated with a problem on uncertainty propagation in a stochastic elliptic partial differential equation (SPDE) with uncertain diffusion coefficient. We deviate from traditional formulations of the SPDE problem by not imposing a specific covariance structure on the random diffusion coefficient. Instead, we attempt to solve a more challenging problem of learning a map between an arbitrary snapshot of the diffusion field and the response.